Nadia Lapusta

Rate and state dependent friction and the stability of sliding between elastically deformable solids

We study the stability of steady sliding between elastically deformable continua using rate and state dependent friction laws. That is done for both elastically identical and elastically dissimilar solids. The focus is on linearized response to perturbations of steady-state sliding, and on studying how the positive direct effect (instantaneous increase or decrease of shear strength in response to a respective instantaneous increase or decrease of slip rate) of those laws allows the existence of a quasi-static range of response to perturbations at sufficiently low slip rate. We discuss the physical basis of rate and state laws, including the likely basis for the direct effect in thermally activated processes allowing creep slippage at asperity contacts, and estimate activation parameters for quartzite and granite. Also, a class of rate and state laws suitable for variable normal stress is presented. As part of the work, we show that compromises from the rate and state framework for describing velocity-weakening friction lead to paradoxical results, like supersonic propagation of slip perturbations, or to ill-posedness, when applied to sliding between elastically deformable solids. The case of sliding between elastically dissimilar solids has the inherently destabilizing feature that spatially inhomogeneous slip leads to an alteration of normal stress, hence of frictional resistance. We show that the rate and state friction laws nevertheless lead to stability of response to sufficiently short wavelength perturbations, at very slow slip rates. Further, for slow sliding between dissimilar solids, we show that there is a critical amplitude of velocity-strengthening above which there is stability to perturbations of all wavelengths.

Elastodynamic Analysis for Slow Tectonic Loading with Spontaneous Rupture Episodes on Faults with Rate- and State-dependent Friction

We present an efficient and rigorous numerical procedure for calculating the elastodynamic response of a fault subjected to slow tectonic loading processes of long duration within which there are episodes of rapid earthquake failure. This is done for a general class of rate- and state-dependent friction laws with positive direct velocity effect. The algorithm allows us to treat accurately, within a single computational procedure, loading intervals of thousands of years and to calculate, for each earthquake episode, initially aseismic accelerating slip prior to dynamic rupture, the rupture propagation itself, rapid post seismic deformation which follows, and also ongoing creep slippage throughout the loading period in velocity-strengthening fault regions. The methodology is presented using the two-dimensional (2-D) antiplane spectral formulation and can be readily extended to the 2-D in-plane and 3-D spectral formulations and, with certain modifications, to the space-time boundary integral formulations as well as to their discretized development using finite difference or finite element methods. The methodology can be used to address a number of important issues, such as fault operation under low overall stress, interaction of dynamic rupture propagation with pore pressure development, patterns of rupture propagation in events nucleated naturally as a part of a sequence, the earthquake nucleation process, earthquake sequences on faults with heterogeneous frictional properties and/or normal stress, and others. The procedure is illustrated for a 2-D crustal strike-slip fault model with depth-variable properties. For lower values of the state-evolution distance of the friction law, small events appear. The nucleation phases of the small and large events are very similar, suggesting that the size of an event is determined by the conditions on the fault segments the event is propagating into rather than by the nucleation process itself. We demonstrate the importance of incorporating slow tectonic loading with elastodynamics by evaluating two simplified approaches, one with the slow tectonic loading but no wave effects and the other with all dynamic effects included but much higher loading rate.

Stable creeping fault segments can become destructive as a result of dynamic weakening

Faults in Earth’s crust accommodate slow relative motion between tectonic plates through either similarly slow slip or fast, seismic-wave-producing rupture events perceived as earthquakes. These types of behaviour are often assumed to be separated in space and to occur on two different types of fault segment: one with stable, rate-strengthening friction and the other with rate-weakening friction that leads to stick-slip. The 2011 Tohoku-Oki earthquake with moment magnitude Mw = 9.0 challenged such assumptions by accumulating its largest seismic slip in the area that had been assumed to be creeping. Here we propose a model in which stable, rate-strengthening behaviour at low slip rates is combined with coseismic weakening due to rapid shear heating of pore fluids, allowing unstable slip to occur in segments that can creep between events. The model parameters are based on laboratory measurements on samples from the fault of the Mw 7.6 1999 Chi-Chi earthquake. The long-term slip behaviour of the model, which we examine using a unique numerical approach that includes all wave effects, reproduces and explains a number of both long-term and coseismic observations—some of them seemingly contradictory—about the faults at which the Tohoku-Oki and Chi-Chi earthquakes occurred, including there being more high-frequency radiation from areas of lower slip, the largest seismic slip in the Tohoku-Oki earthquake having occurred in a potentially creeping segment, the overall pattern of previous events in the area and the complexity of the Tohoku-Oki rupture. The implication that earthquake rupture may break through large portions of creeping segments, which are at present considered to be barriers, requires a re-evaluation of seismic hazard in many areas.

Nucleation and early seismic propagation of small and large events in a crustal earthquake model

[1] Earthquake nucleation and early seismic propagation are studied in a two-dimensional strike-slip fault model with depth-variable properties. The fault is governed by the Dieterich-Ruina rate and state friction law. We use an efficient and rigorous numerical procedure for elastodynamic analysis of earthquake sequences on slowly loaded faults developed by Lapusta et al. [2000]. We find that for decreasing values of the characteristic slip distance of the friction law, small events appear at the transition from the locked to creeping behavior toward the bottom of the seismogenic zone. Small and large events have very similar nucleation phases in our simulations. Here, by ‘‘nucleation phase’’ we mean gradually accelerating aseismic slip in a small slowly expanding zone before the breakout of the dynamic, seismically detectable event. Moment acceleration (to which velocity seismograms are proportional) in early stages of seismic propagation exhibits irregular fluctuations, in the form of speedups and slowdowns in the moment release rate, consistently with observations as reported by Ellsworth and Beroza [1995]. Our simulations show that such irregular moment acceleration can, at least in part, be due to the heterogeneous stress distribution imprinted on the fault by the arrest of previous small events and by stress concentrations at the borders of creeping regions and to partial arrest of the rupture in velocity-strengthening fault regions which inhibit seismic slip. INDEX TERMS: 7209 Seismology: Earthquake dynamics and mechanics; 7260 Seismology: Theory and modeling; 3220 Mathematical Geophysics: Nonlinear dynamics; 7230 Seismology: Seismicity and seismotectonics; 3230 Mathematical Geophysics: Numerical solutions; 8164 Tectonophysics: Stresses—crust and lithosphere; KEYWORDS: earthquake nucleation phase, event clustering, irregular moment release, stress concentrations, rate and state friction, earthquake sequences

Comparison of finite difference and boundary integral solutions to three‐dimensional spontaneous rupture

The spontaneously propagating shear crack on a frictional interface has proven to be a useful idealization of a natural earthquake. The corresponding boundary value problems are nonlinear and usually require computationally intensive numerical methods for their solution. Assessing the convergence and accuracy of the numerical methods is challenging, as we lack appropriate analytical solutions for comparison. As a complement to other methods of assessment, we compare solutions obtained by two independent numerical methods, a finite difference method and a boundary integral (BI) method. The finite difference implementation, called DFM, uses a traction-at-split-node formulation of the fault discontinuity. The BI implementation employs spectral representation of the stress transfer functional. The three-dimensional (3-D) test problem involves spontaneous rupture spreading on a planar interface governed by linear slip-weakening friction that essentially defines a cohesive law. To get a priori understanding of the spatial resolution that would be required in this and similar problems, we review and combine some simple estimates of the cohesive zone sizes which correspond quite well to the sizes observed in simulations. We have assessed agreement between the methods in terms of the RMS differences in rupture time, final slip, and peak slip rate and related these to median and minimum measures of the cohesive zone resolution observed in the numerical solutions. The BI and DFM methods give virtually indistinguishable solutions to the 3-D spontaneous rupture test problem when their grid spacing Δx is small enough so that the solutions adequately resolve the cohesive zone, with at least three points for BI and at least five node points for DFM. Furthermore, grid-dependent differences in the results, for each of the two methods taken separately, decay as a power law in Δx, with the same convergence rate for each method, the calculations apparently converging to a common, grid interval invariant solution. This result provides strong evidence for the accuracy of both methods. In addition, the specific solution presented here, by virtue of being demonstrably grid-independent and consistent between two very different numerical methods, may prove useful for testing new numerical methods for spontaneous rupture problems.

Scaling of small repeating earthquakes explained by interaction of seismic and aseismic slip in a rate and state fault model

Because of short recurrence times and known locations, small repeating earthquakes present a rare predictable opportunity for detailed field observations. They are used to study fault creeping velocities, earthquake nucleation, stress drops, and other aspects of tectonophysics, earthquake mechanics, and seismology. An intriguing observation about repeating earthquakes is their scaling of recurrence time with seismic moment, which is significantly different from the scaling based on a simple conceptual model of circular ruptures with stress drop independent of seismic moment and no aseismic slip. Here we show that a model of repeating earthquakes based on laboratory-derived rate and state friction laws reproduces the observed scaling. In the model, a small fault patch governed by steady state velocity-weakening friction is surrounded by a much larger velocity-strengthening region. Long-term slip behavior of the fault is simulated using a methodology that fully accounts for both aseismic slip and inertial effects of occasional seismic events. The model results in repeating earthquakes with typical stress drops and sizes comparable with observations. For a fixed set of friction parameters, the observed scaling is reproduced by varying the size of the velocity-weakening patch. In simulations, a significant part of slip on the velocity-weakening patches is accumulated aseismically, even though the patches also produce seismic events. The proposed model supplies a laboratory-based framework for interpreting the wealth of observations about repeating earthquakes, provides indirect evidence that rate and state friction acts on natural faults, and has important implications for possible scenarios of slip partition into seismic and aseismic parts.

Three-dimensional boundary integral modeling of spontaneous earthquake sequences and aseismic slip

[1] Fault processes involve complex patterns of seismic events and aseismic slip. This work develops a three-dimensional (3-D) methodology for simulating long-term history of spontaneous seismic and aseismic slip on a vertical planar strike-slip fault subjected to slow tectonic loading. Our approach reproduces all stages of earthquake cycles, from accelerating slip before dynamic instability, to rapid dynamic propagation of earthquake rupture, to postseismic slip, and to interseismic creep, including aseismic transients. We use the developed 3-D methodology to study interaction of fault slip with a small patch of higher normal stress over long-term slip history. For uniform initial prestress, dynamic rupture is significantly affected by the stronger patch in the first simulated event but not in subsequent ones. The change in behavior is due to redistribution of shear stress by prior slip, which demonstrates that distributions of fault strength and stress are related and illustrates the importance of simulating long slip histories even in studies of dynamic rupture. Despite no long-term effect on dynamic rupture, the small patch of higher normal stress influences nucleation processes and hence long-term slip patterns in the model. Comparison of the fully dynamic simulations and a widely used quasi-dynamic approach shows that the quasi-dynamic approach modifies long-term slip patterns in addition to resulting in much smaller slip velocities and rupture speeds during dynamic events. We show that the response of quasi-dynamic formulations with reduced radiation damping terms can be scaled to match the results of the standard quasi-dynamic formulation and hence cannot improve the comparison.

Under the Hood of the Earthquake Machine: Toward Predictive Modeling of the Seismic Cycle

Earthquake Model Shakedown The Parkfield segment of the San Andreas Fault in California experiences Magnitude 6.0 earthquakes at a surprisingly regular interval—roughly every 22 years. This area is one of the most well studied fault segments in the world, yet computational models often struggle to integrate the wealth of observational data with theoretical predictions. Barbot et al. (p. 707; see the Perspective by Segall) constructed a dynamic model of a fault segment which, when integrated with previous observations, reproduces the behavior of the Parkfield segment over the entire earthquake cycle. Because the model is based on realistic fault physics, it not only explains the distribution of small earthquakes but also the recurrence interval of large earthquakes and the amount of geodetic strain accumulated postseismically. It also reveals how smaller earthquakes can influence this regions semiregular earthquake cycle. Computational models predict the long-term recurrence of earthquakes along a segment of the San Andreas Fault. Advances in observational, laboratory, and modeling techniques open the way to the development of physical models of the seismic cycle with potentially predictive power. To explore that possibility, we developed an integrative and fully dynamic model of the Parkfield segment of the San Andreas Fault. The model succeeds in reproducing a realistic earthquake sequence of irregular moment magnitude (Mw) 6.0 main shocks—including events similar to the ones in 1966 and 2004—and provides an excellent match for the detailed interseismic, coseismic, and postseismic observations collected along this fault during the most recent earthquake cycle. Such calibrated physical models provide new ways to assess seismic hazards and forecast seismicity response to perturbations of natural or anthropogenic origins.

Spectral element modeling of spontaneous earthquake rupture on rate and state faults: Effect of velocity-strengthening friction at shallow depths

We develop a spectral-element methodology (SEM) for simulating dynamic rupture on rate and state faults and use it to study how the rupture is affected by a shallow fault region of steady-state velocity-strengthening friction. Our comparison of the developed SEM and a spectral boundary-integral method (BIM) for an anti-plane (two-dimensional) test problem shows that the two methods produce virtually identical solutions for the finest resolution we use and that the convergence with grid reduction of the developed SEM methodology is comparable to that of BIM. We also use the test problem to compare numerical resolution required for different state evolution laws and for linear slip-weakening friction. Using our three-dimensional implementation of the methodology, we find that a shallow velocity-strengthening fault region can significantly alter dynamic rupture and ground motion. The velocity-strengthening region suppresses supershear propagation at the free surface occurring in the absence of such region, which could explain the lack of universally observed supershear rupture near the free surface. In addition, the velocity-strengthening region promotes faster fall-off of slip velocity behind the rupture front and decreases final slip throughout the entire fault, causing a smaller average stress drop. The slip decrease is largest in the shallow parts of the fault, resulting in the depth profile of slip qualitatively consistent with observations of shallow co-seismic slip deficit. The shallow velocity-strengthening region also reduces the amplification of strong ground motion due to a low-velocity bulk structure.

Variability of earthquake nucleation in continuum models of rate-and-state faults and implications for aftershock rates

Using two continuum models of rate-and-state faults, one with a weaker patch and the other with rheological transition from steady state velocity-weakening to velocity-strengthening friction, we simulate several scenarios of spontaneous earthquake nucleation plausible for natural faults, investigate their response to static shear stress steps, and infer the corresponding aftershock rates. Overall, nucleation processes at weaker patches behave similarly to theories based on spring-slider models, with some notable deviations. In particular, nucleation and aftershock rates are affected by normal stress heterogeneity in the nucleation zone. Nucleation processes at rheological transitions behave differently, producing complex slip velocity histories, nonmonotonic responses to static stress changes, and aftershock rates with pronounced peaks and seismic quiescence. For such processes, positive stress steps sometimes delay nucleation of seismic events by inducing aseismic transients that relieve stress and postpone seismic slip. Superposition of the complex aftershock response for spatially variable stress changes results in Omoris law for a period of time followed by seismic quiescence. Such behavior was observed at the base of the seismogenic zone near the 1984 Morgan Hill earthquake. We show that the computed aftershock rates are linked to unperturbed slip velocity evolution in the nucleation zone and construct simplified analytical scenarios that explain some features of the response. The qualitative differences that we find between the two nucleation models indicate that aftershock response of rate-and-state faults to static stress changes would depend on the conditions under which nucleation occurs on natural faults and may be different from predictions based on spring-slider models.